Liouville property, Wiener's test and unavoidable sets for Hunt processes

Abstract

Let $(X,\mathcal W)$ be a balayage space, $1\in \mathcal W$, or - equivalently - let $\mathcal W$ be the set of excessive functions of a Hunt process on a locally compact space $X$ with countable base such that $\mathcal W$ separates points, every function in $\mathcal W$ is the supremum of its continuous minorants and there exist strictly positive continuous $u,v\in \mathcal W$ such that $u/v\to 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $\rho$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property such that $G\approx g\circ\rho$. Assuming that the constant function $1$ is harmonic and balls are relatively compact, is is shown that every positive harmonic function is constant (Liouville property) and that Wiener's test at infinity shows, if a given set $A$ in $X$ is unavoidable, that is, if the process hits $A$ with probability one, wherever it starts. An application yields that locally finite unions of pairwise disjoint balls $B(z,r_z)$, $z\in Z$, which have a certain separation property with respect to a suitable measure $\lambda$ on $X$ are unavoidable if and only if, for some/any point $x_0\in X$, the series $\sum_{z\in Z} g(\rho(x_0,z))/g(r_z) $ diverges. The results generalize and, exploiting a zero-one law for hitting probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and Z. Vondra\v cek, and the author.